Atomic Clocks and Constraints on Variations of Fundamental Constants
Savely G. Karshenboim
D. I. Mendeleev Institute for Metrology (VNIIM), St. Petersburg 198005, Russia
Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
arXiv:physics/0410074v1 [physics.atom-ph] 12 Oct 2004
Victor Flambaum
School of Physics, University of New South Wales, Sydney 2052, Australia
Ekkehard Peik
Physikalisch-Technische Bundesanstalt, 38116 Braunschweig, Germany
(Dated:)
I.
INTRODUCTION
Fundamental constants play an important role in modern physics, being landmarks that designate different
areas. We call them constants, however, as long as
we only consider minor variations with the cosmological
time/space scale, their constancy is an experimental fact
rather than a basic theoretical principle. Modern theories
unifying gravity with electromagnetic, weak, and strong
interactions, or even the developing quantum gravity itself often suggest such variations.
Many parameters that we call fundamental constants,
such as the electron charge and mass (see, e.g., Ref.
[1, 2]), are actually not truly fundamental constants but
effective parameters which are affected by renormalization or the presence of matter [3]. Living in a changing
universe we cannot expect that matter will affect these
parameters the same way during any given cosmological epoch. An example is the inflationary model of the
universe which states that in a very early epoch the universe experienced a phase transition which, in particular,
changed a vacuum average of the so-called Higgs field
which determines the electron mass. The latter was zero
before this transition and reached a value close or equal
to the present value after the transition.
The problem of variations of constants has many facets
and here we discuss aspects related to atomic clocks and
precision frequency measurements. Other related topics
may be found in, e.g., Ref. [4].
Laboratory searches for a possible time variation of
fundamental physical constants currently consist of two
important parts: (i) one has to measure a certain physical quantity at two different moments of time that are
separated by at least a few years; (ii) one has to be able
to interpret the result in terms of fundamental constants.
The latter is a strong requirement for a cross comparison
of different results.
The measurements which may be performed most accurately are frequency measurements; and thus, frequency standards or atomic clocks will be involved in
most of the laboratory searches. Frequency metrology
has shown great progress in the last decade and will continue to do so for some time. The constraints on the variations of the fundamental constants obtained in this man-
ner are, so far, somewhat weaker than those from other
methods (astrophysics, geochemistry), but still competitive with them. In contrast to other methods, however,
frequency measurements allow a very clear interpretation
of the final results and a transparent evaluation procedure, making them less vulnerable to systematic errors.
While there is still potential for improvement, the basic
details of the method have been recently fixed.
The most advanced atomic clocks are discussed in
Sect. II. They are realized with many-electron atoms
and their frequency cannot be interpreted in terms of
fundamental constants. However, a much simpler problem needs to be solved: to interpret their variation in
terms of fundamental constants. This idea is discussed
in Sect. III. The current laboratory constraints on the
variations of the fundamental constants are summarized
in Sect. IV.
II.
ATOMIC CLOCKS AND FREQUENCY
STANDARDS
Frequency standards are important tools for precision
measurements and serve various purposes which, in turn,
have different requirements that must be satisfied. In
particular, it is not necessary for a frequency standard to
reproduce a frequency which is related to a certain atomic
transition although it may be expressed in its terms. A
well known example is the hydrogen maser, where the
frequency is affected by the wall shift which may vary
with time [5]. For the study of time variations of fundamental constants it is necessary to use standards similar
to a primary caesium clock. In this case, any deviation
of its frequency from the unperturbed atomic transition
frequency should be known (within a known uncertainty)
because this is a necessary requirement for being a ‘primary’ standard.
From the point of view of fundamental physics, the
hydrogen maser is an artefact quite similar, in a sense,
to the prototype of the kilogram held at the Bureau International des Poids et Mesures (BIPM) in Paris. Both
artefacts are somehow related to fundamental constants
(e.g., the mass of the prototype can be expressed in terms
of the nucleon masses and their number) but they also
2
have a kind of residual classical-physics flexibility which
allows their properties to change. In contrast, standards
similar to the caesium clock have a frequency (or other
property) that is determined by a certain natural constant which is not flexible, being of pure quantum origin. It may change only if the fundamental constants are
changing.
In Sect. IV, results obtained with caesium and rubidium fountains, a hydrogen beam, ultracold calcium
clouds, and trapped ions of ytterbium and mercury are
discussed. While caesium and rubidium clocks operate in
the radio frequency domain, most of the other standards
listed above rely on optical transitions.
A.
Caesium Atomic Fountain
nance linewidth of 1 Hz is achieved, about a factor of
100 narrower than in traditional devices using a thermal atomic beam from an oven. Selection and detection
of the hyperfine state is performed via optical pumping
and laser induced resonance fluorescence. In a carefully
controlled setup, a relative uncertainty slightly below
1·10−15 can be reached in the realization of the resonance
frequency of the unperturbed Cs atom. The averaging
time that is required to reach this level of uncertainty is
on the order of 104 s. One limiting effect that contributes
significantly to the systematic uncertainty of the caesium
fountain is the frequency shift due to cold collisions between the atoms. In this respect, a fountain frequency
standard based on the ground state hyperfine frequency
of the 87 Rb atom at about 6.835 GHz is more favorable,
since its collisional shift is lower by more than a factor
of 50 for the same atomic density. With the caesium frequency being fixed by definition in the SI system, the
87
Rb frequency is therefore presently the most precisely
measured atomic transition frequency [7].
B.
FIG. 1: Schematic of an atomic fountain clock.
Caesium clocks are the most accurate primary standards for time and frequency [6]. The hyperfine splitting frequency between the F = 3 and F = 4 levels of
the 2 S1/2 ground state of the 133 Cs atom at 9.192 GHz
has been used for the definition of the SI second since
1967. In a so-called caesium fountain (see Fig. 1), a dilute cloud of laser cooled caesium atoms at a temperature of about 1 µK is launched upwards to initiate a
free parabolic flight with an apogee at about 1 m above
the cooling zone. A microwave cavity is mounted near
the lower endpoints of the parabola and is traversed by
the atoms twice – once during ascent, once during descent – so that Ramsey’s method of interrogation with
separated oscillatory fields [5] can be realized. The total
interrogation time being on the order of 0.5 s, a reso-
Single-Ion Trap
An alternative to interrogating atoms in free flight, and
a possibility to obtain practically unlimited interaction
time, is to store them in a trap. Ions are well suited because they carry electric charge and can be trapped in
radio frequency ion traps (Paul traps [8]) that provide
confinement around a field-free saddle point of an electric quadrupole potential. This ensures that the internal
level structure is only minimally perturbed by the trap.
Combined with laser cooling it is possible to reach the
so-called Lamb–Dicke regime where the linear Doppler
shift is eliminated. A single ion, trapped in an ultrahigh
vacuum is conceptually a very simple system that allows
good control of systematic frequency shifts [9]. The use
of the much higher, optical reference frequency allows
one to obtain a stability that is superior to microwave
frequency standards, although only a single ion is used
to obtain a correction signal for the reference oscillator.
A number of possible reference optical transitions with
a natural linewidth of the order of 1 Hz and below are
available in different ions, such as Yb+ [10] and Hg+ [11].
These ions possess a useful level system, where both a
dipole-allowed transition and a forbidden reference transition of the optical clock can be driven with two different lasers from the ground state. The dipole transition
is used for laser cooling and for the optical detection of
the ion via its resonance fluorescence. If a second laser
excites the ion to the metastable upper level of the reference transition, the fluorescence disappears and every
single excitation can thus be detected with practically
hundred percent efficiency as a dark period in the fluorescence signal.
Using these techniques and a femtosecond laser frequency comb generator (see Sect. 99.2.5) for the link to
primary caesium clocks, the absolute frequencies of the
3
ytterbium, or mercury.
metastable
level
cooling
transition
(dipole
allowed)
“forbidden"
reference
transition
FIG. 2: Double resonance scheme applied in single-ion-trap
frequency standards.
transitions 2 S1/2 → 2 D5/2 in 199 Hg+ at 1065 THz and
S1/2 → 2 D3/2 in 171 Yb+ at 688 THz have been measured with relative uncertainties of only 9 · 10−15 . It is
believed that single-ion optical frequency standards offer the potential to ultimately reach the 10−18 level of
relative accuracy.
A similar double resonance technique can be employed
if the reference transition is in the microwave domain and
a number of accurate measurements of hyperfine structure intervals in trapped ions has been performed. In particular, the HFS interval in 171 Yb+ has been measured
several times [12] and can be used to obtain constraints
on temporal variations.
2
C.
Laser-Cooled Neutral Atoms
Optical frequency standards have been developed with
free laser-cooled neutral atoms, most notably of the
alkaline-earth elements that possess narrow intercombination transitions. The atoms are collected in a magnetooptical trap, are then released and interogated by a sequence of laser pulses to realize a frequency-sensitive
Ramsey-Bordé atom interferometer [13]. Of these systems, the one based on the 1 S0 → 3 P1 intercombination
line of 40 Ca at 657 nm has reached the lowest relative uncertainty so far (about 2·10−14 ) [11, 14]. Limiting factors
in the uncertainty of these standards are the residual linear Doppler effect and phase front curvature of the laser
beams that excite the ballistically expanding atom cloud.
It has therefore been proposed to confine the atoms in
an optical lattice, i.e., in the array of interference maxima produced by several intersecting, red-detuned laser
beams [15]. The detuning of the trapping laser could be
chosen such that the light shift it produces in the ground
and excited state of the reference transition are equal,
and therefore it would produce no shift of the reference
frequency. This approach is presently being investigated
and may be applied to the very narrow (mHz natural
linewidth) 1 S0 → 3 P0 transitions in neutral strontium,
D.
Two-Photon Transitions and Doppler-Free
Spectroscopy
The linear Doppler shift of an absorption resonance can
also be avoided if a two-photon excitation is induced by
two counterpropagating laser beams. A prominent example that has been studied with high precision is the twophoton excitation of the 1S → 2S transition in atomic
hydrogen. The precise measurement of this frequency is
of importance for the determination of the Rydberg constant and as a test of quantum electrodynamics (QED).
Hydrogen atoms are cooled by collisions in a cryogenic
nozzle and interact with a standing laser-wave of 243 nm
wavelength inside a resonator. Since the atoms are not
as cold as in laser cooled samples, a correction for the
second order Doppler effect is performed. The laser excitation is interrupted periodically and the excited atoms
are detected in a time resolved manner so that their velocity can be examined. An accuracy of about 2 · 10−14
has been obtained in absolute frequency measurements
with a transportable caesium fountain [16].
E.
Optical Frequency Measurements
In recent years, the progress in stability and accuracy
of optical frequency standards has been impressive; and
there is belief that in the future an optical clock may
supersede the microwave clocks because the optical oscillators offer a much higher number of periods in a given
time. In addition, some systematic effects, such as the
Zeeman effect, have an absolute order of magnitude that
does not scale with the transition frequency, and consequently is relatively less important at higher transition frequencies. A long-standing problem, however, was
the precise conversion of an optical frequency to the microwave domain, where frequencies can be counted electronically in order to establish a time scale or can easily
be compared in a phase coherent way.
This problem has recently been solved by the so-called
femtosecond laser frequency comb generator [17]. Briefly,
a mode-locked femtosecond laser produces, in the frequency domain, a comb of equally spaced optical frequencies fn that can be written as fn = nfr + fceo (with
fceo < fr ), where fr is the pulse repetition rate of the
laser, the mode number n is a large integer (of order
105 ), and fceo (carrier-envelope-offset) is a shift of the
whole comb that is produced by group velocity dispersion in the laser. The repetition rate fr can easily be
measured with a fast photodiode. In order to determine
fceo , the comb is broadened in a nonlinear medium so
that it covers at least one octave. Now the second harmonic of mode n from the “red” wing of the spectrum,
at frequency 2(nfr + fceo ), can be mixed with mode 2n
from the “blue” wing, at frequency 2nfr + fceo , and fceo
4
TABLE I: Limits on possible time variation of frequencies
of different transitions in SI units. Here δf /f is the fractional uncertainty of the most accurate measurement of the
frequency f .
Atom,
transition
H, Opt
Ca, Opt
Rb, HFS
Yb+ , Opt
Yb+ , HFS
Hg+ , Opt
f
δf /f
∆f /∆t
[GHz] [10−15 ]
[Hz/yr]
2 466 061 14
−8 ± 16
455 986
13
−4 ± 5
6.835
1
(0 ± 5) · 10−6
688 359
9
−1 ± 3
12.642
73 (4 ± 4) · 10−4
1 064 721
9
0±7
Refs.
[16]
[14]
[7]
[18]
[12]
[11]
FIG. 3: Frequency comb generated from femtosecond laser
pulses.
is obtained as a difference frequency. In this way, the
precise relation between the two microwave frequencies
fr and fceo and the numerous optical frequencies fn is
known. The setup can now be used for an absolute optical frequency measurement by referencing fr and fceo
to a microwave standard and recording the beat note between the optical frequency fo to be measured and the
closest comb frequency fn . Vice versa, the setup may
work as an optical clockwork, for example, by adjusting
fceo to zero and by stabilizing one comb line fn to fo
so that fr is now an exact subharmonic to order n of
fo . The precision of these transfer schemes has been investigated and was found to be so high that it will not
limit the performance of optical clocks for the foreseeable
future.
F.
The frequency standards described above have been
succesfully developed and their accuracy has been improved in the last decade. This progress, as a consequence, has led to certain constraints on the possible
variations of the fundamental constants. Considering frequency variations, one has to have in mind that not only
the numerical value but also the units may vary. For this
reason, one needs to deal with dimensionless quantities
which are unit-independent. During the last decade, a
number of transition frequencies were measured in the
corresponding SI unit, the hertz. These dimensional results are actually related to dimensionless quantities since
a frequency measurement in SI is a measurement with respect to the caesium hyperfine interval[31]
f
,
fHFS (Cs)
A.
The Spectrum of Hydrogen and Nonrelativistic
Atoms
The hydrogen atom is the simplest atom and one can
easily calculate the leading contribution to different kinds
of transitions in its spectrum (cf., for example, Ref. [19]),
such as the gross, fine, and hyperfine structure. The scaling behavior of these contributions with the values of the
Rydberg constant R∞ , the fine structure constant α, and
the magnetic moments of proton and Bohr magneton is
clear. The results for some typical hydrogenic transitions
are
3
· cR∞ ,
4
1
f (2p3/2 − 2p1/2 ) ≃
· α2 · cR∞ ,
16
4 2 µp
· cR∞ .
fHFS (1s) ≃
·α ·
3
µB
f (2p → 1s) ≃
Limitations on Frequency Variations
f = 9 192 631 770 ·
III. ATOMIC SPECTRA AND THE
FUNDAMENTAL CONSTANTS
(1)
where f stands for the numerical value of the frequency
f . In Sect. IV, in order to simplify notation, this symbol
for the numerical value is dropped.
(2)
In the nonrelativistic approximation, the basic frequencies and the fine and hyperfine structure intervals of all
atomic spectra have a similar dependence on the fundamental constants. The presence of a few electrons and
a nuclear charge of Z 6= 1 makes theory more complicated and introduces certain multiplicative numbers but
involves no new parameters. The importance of this scaling for a search for the variations was first pointed out in
Ref. [20] and was applied to astrophysical data. Similar
results may be presented for molecular transitions (electronic, vibrational, rotational and hyperfine) [21], however, up to now no measurement with molecules has been
performed at a level of accuracy that is competitive with
atomic transitions. They have been used only in a search
for variations of constants in astrophysical observations
(see e.g. [22]).
5
TABLE II: Magnetic moments and relativistic corrections for
atoms involved in microwave standards. The relativistic sensitivity κ is defined in Sect. III C. Here µ is an actual value
of the nuclear magnetic moment, µN is the nuclear magneton, and µS stands for the Schmidt value of the nuclear magnetic moment; the nucleon g factors are gp /2 ≃ 2.79 and
gn /2 ≃ −1.91.
Z Atom
37 87 Rb
55 133 Cs
70 171 Yb+
B.
µ/µN
µS /µN
2.75
gp /2 + 1
2.58 7/18 · (10 − gp )
0.49
−gn /6
µ/µS κ
0.74 0.34
1.50 0.83
0.77 1.5
Hyperfine Structure and the Schmidt Model
µ
· cR∞
µB
(3)
involves nuclear magnetic moments µ which are different for different nuclei; thus, a comparison of the constraints on the variations of nuclear magnetic moments
has a reduced value. To compare them, one may apply
the Schmidt model (see, e.g., Ref. [3, 23]), which predicts
all the magnetic moments of nuclei with an odd number
of nucleons (odd value of atomic number A) in terms of
the proton and neutron g-factors, gp and gn , respectively,
and the nuclear magneton only. Unfortunately, the uncertainty of the calculation within the Schmidt model
is quite high (usually from 10% to 50%). The Schmidt
model, being a kind of ab initio model, only allows for improvements which, unfortunately, involve some effective
phenomenological parameters. This would not really improve the situation, but return us to the case where there
are too many possibly varying independent parameters.
A comparison of the Schmidt values to the actual data
is presented for caesium, rubidium, and ytterbium in Table II.
C.
Atom, transition
∂f /f ∂t
H, 1s − 2s
−3.2(63) × 10−15
40
Ca, 1 S0 −3 P1
−8(11) × 10−15
171
+ 2
2
Yb , S1/2 − D3/2 −1.2(44) × 10−15
199
Hg+ , 2 S1/2 − 2 D5/2 −0.2(70) × 10−15
Atomic Spectra: Relativistic Corrections
A theory based on the leading nonrelativistic approximation may not be accurate enough. Any atomic frequency can be presented as
f = fNR · Frel (α) ,
(4)
where the first (nonrelativistic) factor is determined by a
scaling similar to the hydrogenic transitions (2). The second factor stands for relativistic corrections which vanish
at α = 0; and thus, Frel (0) = 1.
The importance of relativistic corrections for the hyperfine structure was first emphasized in Ref. [25]. Relativistic many-body calculations for various transitions
have been performed in Refs. [24, 26, 27, 28]. A typical
κ
yr−1 0.00
yr−1 0.03
yr−1 0.9
yr−1 −3.2
accuracy is about 10%. Some results are summarized in
Tables II and III, where we list the relative sensitivity of
the relativitic factors Frel to changes in α,
κ=
The atomic hyperfine structure
fNR (HFS) = const · α2 ·
TABLE III: Limits on possible time variation of the frequencies of different transitions and their sensitivity to variations
in α due to relativistic corrections.
∂ ln Frel
.
∂ ln α
(5)
Note that the relativistic corrections in heavy atoms are
proportional to (Zα)2 because of the singularity of relativistic operators. Due to this, the corrections rapidly
increase with the nuclear charge Z.
The signs and magnitudes of κ are explained by a simple estimate of the relativistic correction. For example,
an approximate expression for the relativistic correction
factor for the hyperfine structure of an s-wave electron
in an alkali-like atom is (see, e.g., Ref. [25])
1
11
1
Frel (α) = p
≃ 1 + (Zα)2 .
·
2
2
6
1 − (Zα) 1 − (4/3)(Zα)
A similar rough estimation for the energy levels may be
performed for the gross structure:
Za2 mc2 α2
(Zα)2
1
E=−
.
(6)
· 1+
2n2∗
n∗ j + 1/2
Here j is the electron angular momentum, n∗ is the effective value of the principle quantum number (which determines the nonrelativistic energy of the electron), and
Za is the charge “seen” by the valence electron – it is 1
for neutral atoms, 2 for singly charged ions, etc. This
equation tells us that κ, for the excitation of the electron
from the orbital j to the orbital j ′ , has a different sign
for j > j ′ and j < j ′ . The difference of sign between the
sensitivities of the ytterbium and mercury transitions in
Table III reflects the fact that in Yb+ a 6s-electron is
excited to the empty 5d-shell, while in Hg+ a hole is created in the filled 5d-shell if the electron is excited to the
6s-shell.
IV. LABORATORY CONSTRAINTS ON THE
VARIATIONS OF THE FUNDAMENTAL
CONSTANTS
Logarithmic derivatives [see, e.g., Eq. (5)] appear since
we are looking for a variation of the constants in relative
6
units. In other words, we are interested in a determination of, e.g., ∆α/α∆t while the input data of interest are
related to ∆f /f ∆t. Their relation takes the form
∂ ln fNR
∂ ln α
∂ ln f
=
+κ·
.
∂t
∂t
∂t
(7)
If one compares transitions of the same type – gross structure, fine structure – the first term cancels.
A.
as
2
X 1 ∂ ln fi ∂ ln Ry
∂ ln α
= 1 + χ2min ,
−
− κi
2
u
∂t
∂t
∂t
i
i
is presented in Fig. 5. Here we sum over all available
data: ∂ ln fi /∂t is the central value of the observed drift
rate, ui its 1σ uncertainty, and χ2min the minimized χ2 of
the fit.
Constraints from Absolute Optical
Measurements
Absolute frequency measurements offer the possibility
to compare a number of optical transitions with frequencies fNR , which scale as cR∞ , with the caesium hyperfine
structure. One can rewrite Eq. (7) as
∂ ln fopt
∂ ln cR∞
∂ ln α
=
+κ·
,
∂t
∂t
∂t
(8)
where dimensional quantities, such as frequency and the
Rydberg constant, are stated in SI units [cf. Eq. (1)].
This equation may be used in different ways. For example, in Fig. 4 we plot experimental data for ∂ ln fopt /∂t
as a function of the sensitivity κ and derive a modelindependent constraint on the variation of the fine structure constant
∂ ln α
= (−0.3 ± 2.0) · 10−15 yr−1
∂t
(9)
and the numerical value of the Rydberg frequency cR∞
(see Table IV) in the SI unit of hertz. The latter is of
great metrological importance, being related to a common drift of optical clocks with respect to a caesium
clock, i.e., to the definition of the SI second. The SI definition of the metre is unpractical and so, in practice, the
optical wavelengths of reference lines calibrated against
the caesium standard are used to determine the SI metre
[29].
FIG. 5: Constraints on the time variations of the fine structure constant α and the numerical value of the Rydberg constant. The preliminary data on Ca are not included.
The numerical value of the Rydberg constant, from the
point of view of fundamental physics, can be expressed
in terms of the caesium hyperfine interval in atomic units
and its variation may be expressed in terms of the variations of α and µCs /µB . A constraint for the latter is
presented in Table IV.
B.
Constraints from Microwave Clocks
A model-independent comparison of different HFS
transitions is not simple because their nonrelativistic contributions fNR are not the same, but involve different
magnetic moments. Applying Eq. (9) to experimental
data, one can obtain constraints on the relative variations of the magnetic moments of Rb, Cs, and Yb (see
Table IV).
C.
FIG. 4: Frequency variations versus their sensitivity κ.
The constraints on the variations of α and cR∞ are
correlated and the standard uncertainty ellipse, defined
Model-Dependent Constraints
In order to gain information on constants more fundamental than the nuclear magnetic moments, any further evaluation of the experimental data should involve
the Schmidt model, which is far from perfect. Modeldependent constraints are summarized in Table V.
The nucleon g factors, in their turn, depend on a dimensionless fundamental constant mq /ΛQCD , where mq
is the quark mass and ΛQCD is the quantum chromodynamic (QCD) scale. A study of this dependence may
7
TABLE IV: Model-independent laboratory constraints on the
possible time variations of natural constants.
X
α
{cR∞ }
µCs /µB
µRb /µCs
µYb /µCs
∂ ln X/∂t
(−0.3 ± 2.0) · 10−15 yr−1
(−2.1 ± 3.1) · 10−15 yr−1
(3.0 ± 6.8) · 10−15 yr−1
(−0.2 ± 1.2) · 10−15 yr−1
(3 ± 3) · 10−14 yr−1
TABLE V: Model-dependent laboratory constraints on possible time variations of fundamental constants. The uncertainties here do not include uncertainties from the application of
the Schmidt model.
X
me /mp
µp /µe
gp
gn
∂ ln X/∂t
(2.9 ± 6.2) · 10−15 yr−1
(2.9 ± 5.8) · 10−15 yr−1
(−0.1 ± 0.5) · 10−15 yr−1
(3 ± 3) · 10−14 yr−1
supply us with deep insight into the possible variations of
the more fundamental properties of Nature (see Ref. [24]
for details). This approach is promising, but its accuracy
needs to be better understood.
V.
SUMMARY
for example, not assumed any hierarchy in variation rates
or that some constants stay fixed while others vary, as it is
done in the study of the position of the Oklo resonance.
The evaluation presented here is transparent, and any
particular calculation or measurement can be checked.
In contrast, the astrophysical data show significant results only after an intensive statistical evaluation.
The laboratory searches involving atomic clocks have
definitely shown progress and in a few years we expect
an increase in the accuracy of these clocks, an increase
in the number of different kinds of frequency standards
(e.g., optical Sr, Sr+ , In+ standards and a microwave
Hg+ standard are being tried now), and indeed an increase in the time separation between accurate experiments, since it is now typically only 2–3 years. An optical
clock based on a nuclear transition in Th-229 is also under consideration [30]. Such a clock would offer different
sensitivity to systematic effects, as well as to variations
of different fundamental constants.
Laboratory searches are not necessarily limited by experiments with metrological accuracy. An example of a
high-sensitivity search with a relatively low accuracy is
the study of the dysprosium atom for a determination of
the splitting between the 4f 10 5d6s and 4f 9 5d2 6s states,
which offers a great sensitivity value of κ ≃ 5.7 · 108 [28].
Variations of constants on the cosmological time scale
can be expected but the magnitude, as well as other details, is unclear. Because of a broad range of options
there is a need for the development of as many different
searches as possible, and the laboratory search for variations is an attractive opportunity to open up a way that
could lead to new physics.
The results collected in Tables IV and V are competitive with data from other searches and have a more
reliable interpretation. The results from astrophysical
searches and the study of the samarium resonance from
Oklo data claim higher sensitivity (see, e.g., Ref. [4]),
however, they are more difficult to interpret. We have,
We are very grateful to our colleagues and to participants of the ACFC-2003 meeting for useful and stimulating discussions.
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be published.
[3] S. G. Karshenboim, Eprints physics/0306180 and
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